:: SIN_COS9 semantic presentation

begin

theorem :: SIN_COS9:1
].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) c= dom tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:2
].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) c= dom cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:3
( tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
diff (tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / ((cos : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:4
( cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
diff (cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / ((sin : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:5
tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is continuous ;

theorem :: SIN_COS9:6
cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is continuous ;

theorem :: SIN_COS9:7
tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is increasing ;

theorem :: SIN_COS9:8
cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is decreasing ;

theorem :: SIN_COS9:9
tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is one-to-one ;

theorem :: SIN_COS9:10
cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is one-to-one ;

registration
cluster K77(tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ,].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V19() ) ( V19() Function-like V36() V37() V38() ) set ) -> V19() one-to-one ;
cluster K77(cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ,].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V19() ) ( V19() Function-like V36() V37() V38() ) set ) -> V19() one-to-one ;
end;

definition
func arctan -> ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) equals :: SIN_COS9:def 1
(tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;
func arccot -> ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) equals :: SIN_COS9:def 2
(cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;
end;

definition
let r be ( ( ) ( V11() real ext-real ) Real) ;
func arctan r -> ( ( ) ( ) set ) equals :: SIN_COS9:def 3
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . r : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;
func arccot r -> ( ( ) ( ) set ) equals :: SIN_COS9:def 4
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . r : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;
end;

definition
let r be ( ( ) ( V11() real ext-real ) Real) ;
:: original: arctan
redefine func arctan r -> ( ( ) ( V11() real ext-real ) Real) ;
:: original: arccot
redefine func arccot r -> ( ( ) ( V11() real ext-real ) Real) ;
end;

theorem :: SIN_COS9:11
rng arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) = ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:12
rng arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) = ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

registration
cluster arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) -> Function-like one-to-one ;
cluster arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) -> Function-like one-to-one ;
end;

theorem :: SIN_COS9:13
for x being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( real ) ( V11() real ext-real ) number ) in ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = tan x : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:14
for x being ( ( real ) ( V11() real ext-real ) number ) st x : ( ( real ) ( V11() real ext-real ) number ) in ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = cot x : ( ( real ) ( V11() real ext-real ) number ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:15
for x being ( ( ) ( V11() real ext-real ) Real) st cos : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <> 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = tan x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:16
for x being ( ( ) ( V11() real ext-real ) Real) st sin : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <> 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = cot x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:17
( tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & tan (- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:18
( cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & cot (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . ((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & cot ((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:19
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in [.(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) in [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:20
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in [.(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) in [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:21
rng (tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | [.(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) = [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:22
rng (cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | [.(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) = [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:23
[.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) c= dom arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:24
[.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) c= dom arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

registration
cluster K77(tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ,[.(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V19() ) ( V19() Function-like V36() V37() V38() ) set ) -> V19() one-to-one ;
cluster K77(cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ,[.(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( V19() ) ( V19() Function-like V36() V37() V38() ) set ) -> V19() one-to-one ;
end;

theorem :: SIN_COS9:25
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) = (tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | [.(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:26
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) = (cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | [.(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) " : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:27
(tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | [.(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V19() ) ( V19() Function-like one-to-one V36() V37() V38() ) set ) = id [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:28
(cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | [.(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V19() ) ( V19() Function-like one-to-one V36() V37() V38() ) set ) = id [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:29
(tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | [.(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) = id [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:30
(cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | [.(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) = id [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:31
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * (tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V19() ) ( V19() Function-like one-to-one V36() V37() V38() ) set ) = id ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:32
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * (cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) = id ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:33
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * (tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V19() ) ( V19() Function-like one-to-one V36() V37() V38() ) set ) = id ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:34
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * (cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) | ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( V19() ) ( V19() Function-like one-to-one V36() V37() V38() ) set ) = id ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:35
for r being ( ( ) ( V11() real ext-real ) Real) st - (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) < PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) holds
( arctan (tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Real) = r : ( ( ) ( V11() real ext-real ) Real) & arctan (tan r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Real) = r : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: SIN_COS9:36
for r being ( ( ) ( V11() real ext-real ) Real) st 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) < r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) < PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) holds
( arccot (cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Real) = r : ( ( ) ( V11() real ext-real ) Real) & arccot (cot r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Real) = r : ( ( ) ( V11() real ext-real ) Real) ) ;

theorem :: SIN_COS9:37
( arctan (- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Real) = - (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . (- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:38
( arccot (- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Real) = (3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . (- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:39
( arctan 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Real) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:40
( arccot 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Real) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:41
( tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & tan 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: SIN_COS9:42
( cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & cot (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: SIN_COS9:43
( arctan 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ;

theorem :: SIN_COS9:44
( arccot 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Real) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:45
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | (tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) .: ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is increasing ;

theorem :: SIN_COS9:46
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | (cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) .: ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is decreasing ;

theorem :: SIN_COS9:47
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is increasing ;

theorem :: SIN_COS9:48
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is decreasing ;

theorem :: SIN_COS9:49
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) in [.(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:50
for x being ( ( ) ( ) set ) st x : ( ( ) ( ) set ) in [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( ) set ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) in [.(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:51
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
tan (arctan r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: SIN_COS9:52
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cot (arccot r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) ;

theorem :: SIN_COS9:53
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is continuous ;

theorem :: SIN_COS9:54
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is continuous ;

theorem :: SIN_COS9:55
rng (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) = [.(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:56
rng (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | [.(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) = [.(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,((3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .] : ( ( ) ( closed V46() V47() V48() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:57
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & arctan r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) = - (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) = - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:58
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & arccot r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) = (3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) = - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:59
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & arctan r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SIN_COS9:60
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & arccot r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) = 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SIN_COS9:61
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & arctan r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SIN_COS9:62
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & arccot r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) holds
r : ( ( ) ( V11() real ext-real ) Real) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ;

theorem :: SIN_COS9:63
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( - (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= arctan r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) & arctan r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) <= PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:64
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= arccot r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) & arccot r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) <= (3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:65
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( - (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < arctan r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) & arctan r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) < PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:66
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < arccot r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) & arccot r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) < (3 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 4 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:67
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
arctan r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) = - (arctan (- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:68
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
arccot r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Real) = PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) - (arccot (- r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:69
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
cot (arctan r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:70
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
tan (arccot r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:71
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) is_differentiable_on tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) .: ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:72
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) is_differentiable_on cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) .: ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:73
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) is_differentiable_on ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:74
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) is_differentiable_on ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ;

theorem :: SIN_COS9:75
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
diff (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ,r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (r : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:76
for r being ( ( ) ( V11() real ext-real ) Real) st - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) <= r : ( ( ) ( V11() real ext-real ) Real) & r : ( ( ) ( V11() real ext-real ) Real) <= 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
diff (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ,r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (r : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ;

theorem :: SIN_COS9:77
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | (tan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) .: ].(- (PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,(PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is continuous ;

theorem :: SIN_COS9:78
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) | (cot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) .: ].0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ,PI : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is continuous ;

theorem :: SIN_COS9:79
dom arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) is open ;

theorem :: SIN_COS9:80
dom arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) is open ;

begin

theorem :: SIN_COS9:81
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:82
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:83
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((r : ( ( ) ( V11() real ext-real ) Real) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:84
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( r : ( ( ) ( V11() real ext-real ) Real) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((r : ( ( ) ( V11() real ext-real ) Real) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (r : ( ( ) ( V11() real ext-real ) Real) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:85
for x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) & diff ((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (diff (f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:86
for x being ( ( ) ( V11() real ext-real ) Real)
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) holds
( arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_in x : ( ( ) ( V11() real ext-real ) Real) & diff ((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ,x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - ((diff (f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ,x : ( ( ) ( V11() real ext-real ) Real) )) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ;

theorem :: SIN_COS9:87
for r, s being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + s : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (((r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + s : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:88
for r, s being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + s : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (r : ( ( ) ( V11() real ext-real ) Real) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (((r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + s : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:89
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:90
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:91
for n being ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( (#Z n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * ((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) #Z (n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:92
for n being ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) )
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((#Z n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( (#Z n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((#Z n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - ((n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * ((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) #Z (n : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:93
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) ((#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) ((#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) ((#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:94
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) ((#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) ((#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) ((#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - ((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:95
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + (x : ( ( ) ( V11() real ext-real ) Real) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:96
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) - (x : ( ( ) ( V11() real ext-real ) Real) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:97
for r, s being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + s : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) * (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + (((r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + s : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:98
for r, s being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + s : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (r : ( ( ) ( V11() real ext-real ) Real) * (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) - (((r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + s : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:99
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:100
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:101
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) holds
( f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:102
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:103
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:104
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f1, f2 being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real positive non negative ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:105
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . (x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + (x : ( ( ) ( V11() real ext-real ) Real) / (r : ( ( ) ( V11() real ext-real ) Real) * (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:106
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . (x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) - (x : ( ( ) ( V11() real ext-real ) Real) / (r : ( ( ) ( V11() real ext-real ) Real) * (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:107
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f1, f2, f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = (#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) holds
( f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (r : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:108
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f1, f2, f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) & r : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = (#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) holds
( (r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / (r : ( ( ) ( V11() real ext-real ) Real) * (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:109
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - ((r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & r : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = (#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - ((r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - ((r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . (x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:110
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + ((r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & r : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = (#Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + ((r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) + ((r : ( ( ) ( V11() real ext-real ) Real) / 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . (x : ( ( ) ( V11() real ext-real ) Real) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:111
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:112
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:113
for h, r, s being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + (h : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) + (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) holds
( arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + (h : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + (h : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (s : ( ( ) ( V11() real ext-real ) Real) + ((2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (((r : ( ( ) ( V11() real ext-real ) Real) + (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + (h : ( ( ) ( V11() real ext-real ) Real) * (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:114
for h, r, s being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f, f1, f2 being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + (h : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) + (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) & f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) = #Z 2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) holds
( arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + (h : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * (f1 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) + (h : ( ( ) ( V11() real ext-real ) Real) (#) f2 : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - ((s : ( ( ) ( V11() real ext-real ) Real) + ((2 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) * h : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (((r : ( ( ) ( V11() real ext-real ) Real) + (s : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + (h : ( ( ) ( V11() real ext-real ) Real) * (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:115
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:116
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) holds
( arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - ((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:117
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V11() real ext-real ) Real) * (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:118
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V11() real ext-real ) Real) * (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:119
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:120
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) * arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - ((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:121
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - ((x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:122
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom ((- arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( (- arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
(((- arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - ((x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:123
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = ((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + ((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:124
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = ((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) * (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) - ((exp_R : ( ( Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V33( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) , REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V36() V37() V38() continuous ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:125
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & r : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((((1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (((r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:126
for r being ( ( ) ( V11() real ext-real ) Real)
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) )
for f being ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (((- (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
( f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & r : ( ( ) ( V11() real ext-real ) Real) <> 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) > - 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) & f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) < 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) ) holds
( ((- (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((((- (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / r : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) (#) (arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) * f : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) - (id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = - (((r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + ((r : ( ( ) ( V11() real ext-real ) Real) * x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:127
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = ((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + ((ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:128
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = ((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) - ((ln : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:129
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (- ((arctan : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) + (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V11() real ext-real ) Real) * (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;

theorem :: SIN_COS9:130
for Z being ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) st not 0 : ( ( ) ( natural V11() real ext-real V46() V47() V48() V49() V50() V51() V63() V64() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= dom (((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) : ( ( ) ( V46() V47() V48() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) & Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) c= ].(- 1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) ) : ( ( ) ( non empty V11() real ext-real non positive negative V63() ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ,1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) .[ : ( ( ) ( open V46() V47() V48() V82() V83() V87() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) holds
( ((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) is_differentiable_on Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) & ( for x being ( ( ) ( V11() real ext-real ) Real) st x : ( ( ) ( V11() real ext-real ) Real) in Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) holds
((((id Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) ^) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) (#) arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) `| Z : ( ( open ) ( open V46() V47() V48() ) Subset of ( ( ) ( ) set ) ) ) : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like V36() V37() V38() ) Element of K6(K7(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ,REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V36() V37() V38() ) set ) ) : ( ( ) ( ) set ) ) . x : ( ( ) ( V11() real ext-real ) Real) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) = (- ((arccot : ( ( Function-like ) ( V19() V22( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) V23( REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) Function-like one-to-one V36() V37() V38() ) PartFunc of ,) . x : ( ( ) ( V11() real ext-real ) Real) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) / (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) - (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) / (x : ( ( ) ( V11() real ext-real ) Real) * (1 : ( ( ) ( non empty natural V11() real ext-real positive non negative V46() V47() V48() V49() V50() V51() V63() V64() V82() V84() ) Element of NAT : ( ( ) ( V46() V47() V48() V49() V50() V51() V52() V84() ) Element of K6(REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( ) set ) ) ) + (x : ( ( ) ( V11() real ext-real ) Real) ^2) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) : ( ( ) ( V11() real ext-real ) Element of REAL : ( ( ) ( non empty V46() V47() V48() V52() V73() V84() V85() V87() ) set ) ) ) ) ;